Examining finite-time behaviors in the fractional Gray–Scott model: Stability, synchronization, and simulation analysis

This paper investigates the behavior and stability of the fractional-order Gray–Scott model, with a specific focus on achieving finite-time stability and synchronization. It introduces essential concepts, including the Gamma function, the Riemann–Liouville fractional-order integral operator, the Caputo fractional derivative, and the Mittag-Leffler function, to establish a foundational framework for subsequent analysis. Equilibrium points are defined, distinguishing between initial and finite-time equilibria, and the conditions for finite-time stability, including settling time, are precisely outlined. Stability results for this model are presented through theorems with detailed proofs, elucidating the roles of Lyapunov functions, class functions, and other system parameters. Furthermore, the paper explores finite-time synchronization schemes in master–slave systems, providing a mathematical framework for understanding and achieving synchronization within a finite time frame. This framework illuminates synchronization dynamics and their practical implications for controlling complex systems. Additionally, numerical examples illustrate finite-time stability and synchronization within the Gray–Scott reaction–diffusion model.